Now we shall explain known results for the Schrödinger operator on \({\mathbb {R}}\)  more precisely. Weder [1]} proved \(W_\pm \in \mathbb {B}(L^p({\mathbb {R}}))\)  for \(1<p<\infty \)  if \({\langle }x{\rangle }^{\gamma }V\in L^1({\mathbb {R}})\)  with some \(\gamma >3/2\)  in the regular case and \(\gamma >5/2\)  in the zero resonant case. Artbazar–Yajima [2]} also proved independently a similar result under a slightly stronger decay condition on \(V\) . Later, the assumption on \(V\)  has been weakened to \({\langle }x{\rangle }V\in L^1({\mathbb {R}})\)  in the regular case and \({\langle }x{\rangle }^2V\in L^1({\mathbb {R}})\)  in the zero resonant case by D'Ancona–Fanelli [3]}, and finally to \({\langle }x{\rangle }V\in L^1({\mathbb {R}})\)  in the zero resonant case by Weder [4]}. It was also shown by [1]} that \(W_\pm \in \mathbb {B}(W^{k,p}({\mathbb {R}}))\cap \mathbb {B}(L^1({\mathbb {R}}),L^{1,\infty }({\mathbb {R}}))\cap \mathbb {B}({\mathcal {H}}^1({\mathbb {R}}),L^{1}({\mathbb {R}}))\)  for general cases and that \(W_\pm \in \mathbb {B}(L^1({\mathbb {R}}))\cap \mathbb {B}(L^\infty ({\mathbb {R}}))\)  if zero is a resonance and the scattering matrix at \(\lambda =0\)  is the identity matrix. It was also mentioned in [1]} that \(W_\pm \)  are neither bounded on \(L^1({\mathbb {R}})\)  nor on \(L^\infty ({\mathbb {R}})\)  in general. The case with a delta potential \(V=a\delta \)  was studied by Duchêne–Marzuola–Weinstein [7]} and Weder [4]}. Weder [4]} also studied the case with matrix Schrödinger operators on the line or the half line. Note that, in all these papers [2]}, [3]}, [7]}, [1]}, [4]}, the proofs heavily rely on the Jost functions and their properties studied by Deift–Trubowitz [15]}.
